The construction of \(\alpha ^k\)  derivation and a representation theory were investigated [1]}. Also, the cohomology complex of Hom-Lie superalgebras was furnished and the central extensions was computed. As application, the derivations and the second cohomology group of a twisted \(osp(1,2)\)  superalgebra were calculated. Moreover, Curtright and Zachos introduced the \(q-\)  deformed Witt algebra [2]} and from this results Ding et al  determined a nontrivial \(q-\)  deformed Witt \(n-\)  algebras. It is a generalization of the Lie algebra also called sh-n-Lie algebra [3]}.
Wang et al [4]}
investigated the two different \(q-\)  deformed Witt algebra and constructed their \(n-\)  algebras. In
one case, the super version is also presented. Moreover the central
extensions is provided and the super \(q-\)  deformed Virasoro \(n-\)  algebra for the \(n\)  even case is furnished.
